The problem of allocating divisible goods has enjoyed a lot of attention in both mathematics (e.g. the cake-cutting problem) and economics (e.g. market equilibria). On the other hand, the natural requirement of indivisible goods has been somewhat neglected, perhaps because of its more complicated nature. In this work we study a fairness criterion, called the Max-Min Fairness problem, for k players who want to allocate among themselves m indivisible goods. Each player has a specified valuation function on the subsets of the goods and the goal is to split the goods between the players so as to maximize the minimum valuation. Viewing the problem from a game theoretic perspective, we show that for two players and additive valuations the expected minimum of the (randomized) cut-and-choose mechanism is a 1/2-approximation of the optimum. To complement this result we show that no truthful mechanism can compute the exact optimum.We also consider the algorithmic perspective when the (true) additive valuation functions are part of the input. We present a simple 1/( m - k + 1) approximation algorithm which allocates to every player at least 1/ k fraction of the value of all but the k - 1 heaviest items. We also give an algorithm with additive error against the fractional optimum bounded by the value of the largest item. The two approximation algorithms are incomparable in the sense that there exist instances when one outperforms the other.

We present an improved "cooling schedule" for simulated annealing algorithms for combinatorial counting problems. Under our new schedule the rate of cooling accelerates as the temperature decreases. Thus, fewer intermediate temperatures are needed as the simulated annealing algorithm moves from the high temperature (easy region) to the low temperature (difficult region). We present applications of our technique to colorings and the permanent (perfect matchings of bipartite graphs). Moreover, for the permanent, we improve the analysis of the Markov chain underlying the simulated annealing algorithm. This improved analysis, combined with the faster cooling schedule, results in an O(n 7 log 4 n) time algorithm for approximating the permanent of a 0/1 matrix.

We study the complexity of approximating the value of the independent set polynomial Z G ( ) of a graph G with maximum degree when the activity is a complex number.When is real, the complexity picture is well-understood, and is captured by two real-valued thresholds ⇤ and c , which depend on and satisfy 0 < ⇤ < c . It is known that if is a real number in the interval ( ⇤ , c ) then there is an FPTAS for approximating Z G ( ) on graphs G with maximum degree at most . On the other hand, if is a real number outside of the (closed) interval, then approximation is NP-hard. The key to establishing this picture was the interpretation of the thresholds ⇤ and c on the -regular tree. The "occupation ratio" of a -regular tree T is the contribution to Z T ( ) from independent sets containing the root of the tree, divided by Z T ( ) itself. This occupation ratio converges to a limit, as the height of the tree grows, if and only if 2Unsurprisingly, the case where is complex is more challenging. It is known that there is an FPTAS when is a complex number with norm at most ⇤ and also when is in a small strip surrounding the real interval [0, c ). However, neither of these results is believed to fully capture the truth about when approximation is possible. Peters and Regts identified the values of for which the occupation ratio of the -regular tree converges. These values carve a cardioid-shaped region ⇤ in the complex plane, whose boundary includes the critical points ⇤ and c . Motivated by the picture in the real case, they asked whether ⇤ marks the true approximability threshold for general complex values .Our main result shows that for every outside of ⇤ , the problem of approximating Z G ( ) on graphs G with maximum degree at most is indeed NP-hard. In fact, when is outside of ⇤ and is not a positive real number, we give the stronger result that approximating Z G ( ) is actually #P-hard. Further, on the negative real axis, when < ⇤ , we show that it is #P-hard to even decide whether Z G ( ) > 0, resolving in the affirmative a conjecture of Harvey, Srivastava and Vondrák.Our proof techniques are based around tools from complex analysis -specifically the study of iterative multivariate rational maps. The full version is available at arxiv.org/abs/1711.00282 and is attached as an appendix. The theorem numbering here matches the full version.

Approximate counting via correlation decay is the core algorithmic technique used in the sharp delineation of the computational phase transition that arises in the approximation of the partition function of anti-ferromagnetic two-spin models.Previous analyses of correlation-decay algorithms implicitly depended on the occurrence of strong spatial mixing. This, roughly, means that one uses worst-case analysis of the recursive procedure that creates the sub-instances. In this paper, we develop a new analysis method that is more refined than the worst-case analysis. We take the shape of instances in the computation tree into consideration and we amortise against certain "bad" instances that are created as the recursion proceeds. This enables us to show correlation decay and to obtain an FPTAS even when strong spatial mixing fails.We apply our technique to the problem of approximately counting independent sets in hypergraphs with degree upper-bound ∆ and with a lower bound k on the arity of hyperedges. Liu and Lin gave an FPTAS for k ≥ 2 and ∆ ≤ 5 (lack of strong spatial mixing was the obstacle preventing this algorithm from being generalised to ∆ = 6). Our technique gives a tight result for ∆ = 6, showing that there is an FPTAS for k ≥ 3 and ∆ ≤ 6. The best previously-known approximation scheme for ∆ = 6 is the Markov-chain simulation based FPRAS of Bordewich, Dyer and Karpinski, which only works for k ≥ 8.Our technique also applies for larger values of k, giving an FPTAS for k ≥ ∆. This bound is not substantially stronger than existing randomised results in the literature. Nevertheless, it gives the first deterministic approximation scheme in this regime. Moreover, unlike existing results, it leads to an FPTAS for counting dominating sets in regular graphs with sufficiently large degree.We further demonstrate that in the hypergraph independent set model, approximating the partition function is NP-hard even within the uniqueness regime. Also, approximately

We study the problem of counting and randomly sampling binary contingency tables. For given row and column sums, we are interested in approximately counting (or sampling) 0/1 n × m matrices with the specified row/column sums. We present a simulated annealing algorithm with running time O((nm) 2 D 3 d max log 5 (n + m)) for any row/column sums where D is the number of non-zero entries and d max is the maximum row/column sum. In the worst case, the running time of the algorithm is O(n 11 log 5 n) for an n × n matrix. This is the first algorithm to directly solve binary contingency tables for all row/column sums. Previous work reduced the problem to the permanent, or restricted attention to row/column sums that are close to regular. The interesting aspect of our simulated annealing algorithm is that it starts at a non-trivial instance, whose solution relies on the existence of short alternating paths in the graph constructed by a particular Greedy algorithm.